Variances and central limit theorems for random beta-polytopes and in other geometric models

TL;DR

This paper establishes asymptotic bounds and central limit theorems for variances and distributions of geometric features in various random polytopes across Euclidean, spherical, and hyperbolic spaces.

Contribution

It provides the first comprehensive variance bounds and CLTs for intrinsic volumes and face counts in multiple geometric models, extending previous results.

Findings

Matching asymptotic bounds on variances of intrinsic volumes and face counts.

Central limit theorems for intrinsic volumes using Stein's method.

Asymptotic upper bounds on variances of volume and vertices in spherical and hyperbolic polytopes.

Abstract

We prove matching asymptotic lower and upper bounds on the variances of the intrinsic volumes and the number of $k$-faces of $d$-dimensional random beta-polytopes. Using Stein's methods, we establish central limit theorems for the intrinsic volumes. We also prove asymptotic upper bounds on the variances of the volume and vertex number of spherical random polytopes in spherical convex bodies, and hyperbolic random polytopes in convex bodies in hyperbolic space. Moreover, we consider a circumscribed model on the sphere.

Equations

$$f_{d,\beta }(x)=c_{d,\beta }(1-\Vert x{\Vert }^2{)}^{\beta },\Vert x\Vert <1,\text{ with }{c}_{d,\beta }=\frac{\Gamma (\frac{d}{2}+\beta +1)}{{\pi }^{\frac{d}{2}}\Gamma (\beta +1)},$$

$${\mathrm{Vol}}_d(K+\varrho B^d)=\sum _{s=0}^d{\varrho }^{d-s}{\kappa }_{d-s}{V}_s(K),$$

$$\mathrm{Var}{V}_s(K_n^{\beta })\approx n^{-\frac{d+3}{d+1+2\beta }}.$$

$$d_W(X,Y):=\underset{h\in {\text{Lip}}_1}{\sup }|\mathbb{E}h(X)-\mathbb{E}h(Y)|,$$

$$d_W(\frac{V_s(K_n^{\beta })-\mathbb{E}{V}_s(K_n^{\beta })}{\sqrt{\mathrm{Var}{V}_s(K_n^{\beta })}},G)\ll n^{-\frac{1}{2}+\frac{1+\beta }{d+1+2\beta }}(\log n{)}^{\frac{3(d+1)+2+2\beta }{d+1+2\beta }},$$

$$d_W(\frac{V_s(K_n^{\beta })-\mathbb{E}{V}_s(K_n^{\beta })}{\sqrt{\mathrm{Var}{V}_s(K_n^{\beta })}},G)\ll n^{-\frac{1}{2}+\frac{1+\beta }{d+1+2\beta }}(\log n{)}^{\frac{3(d+1)+1+3\beta }{d+1+2\beta }}.$$

$$\frac{V_s(K_n^{\beta })-\mathbb{E}{V}_s(K_n^{\beta })}{\sqrt{\mathrm{Var}{V}_s(K_n^{\beta })}}\stackrel{d}{\to }G,$$

$$\mathrm{Var}{f}_k(K_n^{\beta })\approx n^{\frac{d-1}{d+1+2\beta }}.$$

$$\underset{n\to \infty }{\lim }\mathbb{E}({\mathrm{Vol}}_{{\mathcal{M}}^d}(K\setminus K_n))\cdot n^{\frac{2}{d+1}}={\gamma }_d{\mathrm{Vol}}_{{\mathcal{M}}^d}(K{)}^{\frac{2}{d+1}}{\int }_{\partial K}{H}_{d-1}^{{\mathcal{M}}^d}(K,x{)}^{\frac{1}{d+1}}\mathrm{d}x,$$

$$\underset{n\to \infty }{\lim }\mathbb{E}(f_0(K_n))\cdot n^{-\frac{d-1}{d+1}}={\gamma }_d{\mathrm{Vol}}_{{\mathcal{M}}^d}(K{)}^{-\frac{d-1}{d+1}}{\int }_{\partial K}{H}_{d-1}^{{\mathcal{M}}^d}(K,x{)}^{\frac{1}{d+1}}\mathrm{d}x.$$

$$\mathrm{Var}{\mathrm{Vol}}_{{\mathcal{M}}^d}(K_n)$$

$$\mathrm{Var}{f}_0(K_n)$$

$$\underset{n\to \infty }{\lim }({\mathrm{Vol}}_{{\mathcal{M}}^d}(K\setminus K_n))\cdot n^{\frac{2}{d+1}}={\gamma }_d{\mathrm{Vol}}_{{\mathcal{M}}^d}(K{)}^{\frac{2}{d+1}}{\int }_{\partial K}{H}_{d-1}^{{\mathcal{M}}^d}(K,x{)}^{\frac{1}{d+1}}\mathrm{d}x.$$

$$\underset{n\to \infty }{\lim }{f}_0(K_n)\cdot n^{-\frac{d-1}{d+1}}={\gamma }_d{\mathrm{Vol}}_{{\mathcal{M}}^d}(K{)}^{-\frac{d-1}{d+1}}{\int }_{\partial K}{H}_{d-1}^{{\mathcal{M}}^d}(K,x{)}^{\frac{1}{d+1}}\mathrm{d}x.$$

$$\mathrm{Var}{\mathrm{Vol}}_{{\mathcal{M}}^d}(K_n)\approx n^{-\frac{d+3}{d+1}}.$$

$$\mu (A)=\frac{1}{{\omega }_{d+1}}{\int }_{{\mathbb{S}}^d}1(H^{-}(x)\in A)\mathrm{d}x,$$

$$U_1(K)=\frac{1}{2}{\int }_{G(d+1,d)}\chi (K\cap H)\mathrm{d}\nu (H),$$

$$\underset{n\to \infty }{\lim }{\mathbb{E}}_{{\mu }_K}(U_1(K^{(n)})-U_1(K))\cdot n^{\frac{2}{n+1}}=\frac{{\gamma }_d}{{\omega }_{d+1}}{\mathrm{Vol}}_{{\mathbb{S}}^d}(K^{\ast }{)}^{\frac{2}{d+1}}{\int }_{\partial K}{H}_{d-1}^{{\mathbb{S}}^d}(K,x{)}^{\frac{d}{d+1}}\mathrm{d}x,$$

$$\underset{n\to \infty }{\lim }{\mathbb{E}}_{{\mu }_K}{f}_{d-1}(K^{(n)})\cdot n^{-\frac{d-1}{n+1}}={\gamma }_d{\mathrm{Vol}}_{{\mathbb{S}}^d}(K^{\ast }{)}^{-\frac{d-1}{d+1}}{\int }_{\partial K}{H}_{d-1}^{{\mathbb{S}}^d}(K,x{)}^{\frac{d}{d+1}}\mathrm{d}x.$$

$$\mathrm{Var}{U}_1(K^{(n)})$$

$$\mathrm{Var}{f}_{d-1}(K^{(n)})$$

$$K[v]=\bigcap \{H^{-}:{\mathrm{Vol}}_{{\mathcal{M}}^d}(K\cap H^{+})\le v\}.$$

$$\underset{n\to \infty }{\lim }{\mathbb{E}}_{\varrho }(V_{\lambda }(K_{(n)}))\cdot n^{\frac{2}{d+1}}$$

$$\underset{n\to \infty }{\lim }{\mathbb{E}}_{\varrho }(f_0(K_{(n)}))\cdot n^{-\frac{d-1}{d+1}}$$

$$\mathrm{Var}{}_{\varrho }(V_{\lambda }(K_{(n)}))\gg n^{-\frac{d+3}{d+1}},\frac{V_{\lambda }(K_{(n)})-{\mathbb{E}}_{\varrho }{V}_{\lambda }(K_{(n)})}{\sqrt{{\mathrm{Var}}_{\varrho }{V}_{\lambda }(K_{(n)})}}\stackrel{d}{\to }G,$$

$$\mathrm{Var}{}_{\varrho }(V_{\lambda }(K_{(n)}))$$

$$\mathrm{Var}{}_{\varrho }(f_0(K_{(n)}))$$

$$g(x)=\frac{x}{⟨x,e_{d+1}⟩}-e_{d+1},$$

$$h(x)=\left(\frac{x_1}{x_{d+1}},\dots ,\frac{x_d}{x_{d+1}}\right).$$

$${\mathbb{P}}_{\beta }(C(t))\approx t^{\frac{d+1+2\beta }{2}},\text{ as }t\to 0^{+}.$$

$$F_{1,\beta }(h)$$

$$B(a,b)$$

$$B(z;a,b)$$

$$F_{1,\beta }$$

Full text

1. Introduction and results

The study of random polytopes generated as the convex hull of random points is a highly diverse field of geometry. The number and distribution of the random points, the different concepts of convexity, and the various geometric functionals considered all represent distinct models. For a general overview, detailed history, and references of the extensive literature of this beautiful and rapidly developing topic, see, for example, the surveys by Bárány [Ba08], Reitzner [R10], Schneider [Sch18], Weil and Wieacker [WW93], and the book by Schneider and Weil [SW08].

In this paper, we focus on rotationally symmetric distributions, such as beta-type distributions in $\mathbb{R}^{d}$ and the uniform distributions on the unit sphere $\mathbb{S}^{d}$ and in hyperbolic space $\mathbb{H}^{d}$.

Random beta-polytopes

First, we consider the so-called beta-distributions in $\mathbb{R}^{d}$. For $\beta>-1$, we say that a random point in the interior of the $d$-dimensional unit ball $B^{d}$ is beta-distributed if its probability density function is

[TABLE]

where $\|\cdot\|$ denotes the Euclidean norm and $\Gamma$ is Euler’s gamma function.

In [KTT19], some special cases are highlighted: $\beta=0$ yields the uniform model in which the random points are uniformly distributed in the unit ball; the uniform distribution on the sphere $\mathbb{S}^{d-1}$ is the weak limit of the beta-distribution as $\beta\to-1$; and suitably adjusting the beta-distribution leads to the Gaussian distribution.

The asymptotics of the expectation of various geometric quantities, as the number $n$ of points tends to infinity, have been studied, for instance, by Affentranger [A91], Carnal [C70], Dwyer [D91], Eddy and Gale [EG81], and Hashorva [H11]. For further references and history on random beta-polytopes, see, for instance, [K23], [KP25], [KS25], [KTT19], [M71], and the references therein.

In this paper, we consider the following model. Let $x_{1},\ldots,x_{n}$ be i.i.d. random points chosen in $B^{d}$ according to the beta-distribution for a fixed $\beta>-1$. Denote the convex hull of a set of points $P\subset\mathbb{R}^{d}$ by $[P]$. The convex hull $K_{n}^{\beta}=[x_{1},\ldots,x_{n}]$ is called a random beta-polytope. The exact expectations of the number of facets and the intrinsic volumes have been determined by Kabluchko, Temesvari, and Thäle in [KTT19], among other results. We recall that the intrinsic volumes $V_{s}(K)$, $s=0,\ldots,d$, of a convex body (compact, convex set with nonempty interior) $K$ are defined as the coefficients of the Steiner formula

[TABLE]

where $K+\varrho B^{d}$ is the Minkowski sum of $K$ and the ball of radius $\varrho\geq 0$, and $\kappa_{d}$ denotes the volume of the $d$-dimensional unit ball. In particular, $V_{d}(K)=\operatorname{Vol}_{d}(K)$ is the $d$-dimensional volume (Lebesgue measure), $V_{0}(K)=1$, $V_{1}$ is a constant times the mean width, and $V_{d-1}$ is half of the surface area. For the $s$-dimensional volume ($s\leq d$), we will use $\operatorname{Vol}_{s}$.

To the best of our knowledge, asymptotic results for the variances of the intrinsic volumes and central limit theorems are not known for general $\beta>-1$ (cf. [KS25]*Remark 1.4), only in the uniform case, i.e., $\beta=0$.

When the i.i.d. random points are chosen according to the uniform distribution in $B^{d}$, Küfer [Kuf94] proved upper bound for the variance of the volume of the random polytope. Reitzner [R03],[R05] proved lower and upper bounds with matching order of magnitude for the variance of the volume when the uniform random points are chosen in a convex body $K$ with $C^{2}_{+}$ smooth boundary. Böröczky, Fodor, Reitzner and Vígh [BFRV09] proved an upper bound on the variance of the mean width of the random polytope in the case when $K$ has a rolling ball, and for convex bodies with $C^{2}_{+}$ smooth boundary, Bárány, Fodor, and Vígh [BFV10] proved matching lower and upper bounds for the order of magnitude of the variance of the intrinsic volumes of the random polytope. Thäle, Turchi and Wespi [TTW18] proved central limit theorems for the intrinsic volumes. For the number of $k$-dimensional faces $f_{k}$, ($k\in\{0,1,\ldots,d-1\}$), Reitzner [R05], [R05b] proved asymptotic lower and upper bounds on the variance with matching orders of magnitude.

For real functions $f$ and $g$ that are defined on the same space $I$, we write $f\ll g$ if there exists a constant $\gamma>0$ such that $|f(x)|\leq\gamma g(x)$ for all $x\in I$. If $f\ll g$ and $g\ll f$, we write $f\approx g$.

One of our main results is the following theorem, which determines the order of magnitude of ${\rm Var}\,V_{s}(K_{n}^{\beta})$.

Theorem 1**.**

For $s=1,\ldots,d$,

[TABLE]

The upper bound in Theorem 1 implies a strong law of large numbers, i.e., for $s=1,\ldots,d$, $V_{s}(K_{n}^{\beta})$ almost surely tends to its expected value, as $n\to\infty$.

The lower bound on the variances can be used to prove central limit theorems for $V_{s}(K_{n}^{\beta})$. Let $d_{W}(\cdot,\cdot)$ denote the Wasserstein distance of two random variables, defined as

[TABLE]

where the supremum is taken over all Lipschitz functions $h:\mathbb{R}\to\mathbb{R}$ with Lipschitz constant at most $1$. We use the fact that if $G$ is a standard Gaussian random variable and $(W_{n})_{n\in\mathbb{N}}$ is a sequence of centered random variables with finite second moments for which $d_{W}(W_{n}/\sqrt{{\rm Var}\,W_{n}},G)\to 0$, as $n\to\infty$, then $W_{n}/\sqrt{{\rm Var}\,W_{n}}$ converges in distribution to $G$.

Similarly to [TTW18], we use the normal approximation bound based on Stein’s method [Ste86], which had been used by, for instance, Besau and Thäle [BT20], Chatterjee [Cha08], Fodor and Papvári [FP24], and Lachièze-Rey and Peccati [LRP17], for different models.

In this paper, we prove the following central limit theorems on $V_{s}(K_{n}^{\beta})$ for $s=1,\ldots,d$ and $\beta>-1$.

Theorem 2**.**

Let $G$ be a standard Gaussian random variable. Then, for $s=1,\ldots,d$, depending on the value of $\beta$, the following hold.

1. (i)

For $-1<\beta\leq 1$,

[TABLE] 2. (ii)

and for $\beta>1$,

[TABLE]

Notice that the exponent of $n$ in both parts of Theorem 2 is negative for any $\beta>-1$ and $d\geq 2$, so the right-hand sides tend to zero as $n\to\infty$, and therefore,

[TABLE]

where $\xrightarrow{d}$ means convergence in distribution. However, the rate of convergence depends on $\beta$, and the function in the exponent of the logarithm changes at $\beta=1$.

We note that in the special case of $\beta=0$, Theorem 2 gives back the results of [TTW18] in the uniform model. The limit of the right-hand side of part $(i)$ of Theorem 2, as $\beta\to-1$, gives back the result of Thäle [Th18], where the random points are uniformly distributed on the boundary of a convex body with a smooth boundary.

We also prove matching asymptotic lower and upper bounds for the number of $k$-dimensional faces $f_{k}$, $k\in\{0,1,\ldots,d-1\}$, of the random beta polytope. For the expectation of the $f$-vector, exact formulas for finite $n$, as well as asymptotics as $n\to\infty$, have been proved by Kabluchko, Thäle and Zaporozhets [KTZ20].

Theorem 3**.**

For $k=0,1,\ldots,d-1$,

[TABLE]

The order of magnitude in Theorem 3 coincides with the uniform model for $\beta=0$, and for $d\geq 4$, the limit as $\beta\to-1$ matches with [RS25].

Non-Euclidean random polytopes

Random polytopes in non-Euclidean geometries have recently attracted much attention. It is a natural problem to try to transfer statements from Euclidean theory to more general settings. For an overview of such results, see, for example, Besau, Ludwig, and Werner [BLW18], Besau and Thäle [BT20], Kabluchko and Panzo [KP25], Schneider [Sch22], and the references therein.

For $d\geq 2$, let $\mathcal{M}^{d}$ denote one of the following spaces: Euclidean $d$-space $\mathbb{R}^{d}$, the unit sphere $\mathbb{S}^{d}$ in $\mathbb{R}^{d+1}$, or the hyperbolic space $\mathbb{H}^{d}=\{x\in\mathbb{R}^{d+1}:x_{1}^{2}+\ldots x_{d}^{2}-x_{d+1}^{2}=-1,\;x_{d+1}>0\}$. Endow each space with the corresponding geodesic distance: the metric $d$ induced by the Euclidean scalar product $\langle\cdot,\cdot\rangle$ in $\mathbb{R}^{d}$, $d_{S}(x,y)=\arccos\langle x,y\rangle$ for $x,y\in\mathbb{S}^{d}$, and $\cosh d_{H}(x,y)=x_{d+1}y_{d+1}-x_{1}y_{1}-\ldots-x_{d}y_{d}$ for $x,y\in\mathbb{H}^{d}$.

We call a set $K$ convex in $\mathcal{M}^{d}$ if, together with any two of its points, the unique geodesic segment connecting them is also contained in $K$. A convex body is a compact convex set in $\mathcal{M}^{d}$ with non-empty interior. We note that a spherical convex body is always contained in an open hemisphere.

The volume (Lebesgue measure) of a measurable set in $\mathcal{M}^{d}$ is denoted by $\operatorname{Vol}_{\mathcal{M}^{d}}(\cdot)$, or specifically $\operatorname{Vol}_{d}(\cdot)$ in $\mathbb{R}^{d}$. Let $\mathcal{K}(\mathcal{M}^{d})$ denote the family of convex bodies in $\mathcal{M}^{d}$. The convex hull of a closed set $X\subset\mathcal{M}^{d}$ is the intersection of all closed half-spaces containing $X$ if $\mathcal{M}^{d}=\mathbb{R}^{d}$ or $\mathbb{H}^{d}$, and the intersection of all closed hemispheres containing $X$ if $\mathcal{M}^{d}=\mathbb{S}^{d}$.

In this paper, we consider the following probabilistic model for random polytopes. Let $K\in\mathcal{K}(\mathcal{M}^{d})$, and let $x_{1},\ldots,x_{n}$ be $n$ independent random points chosen according to the uniform distribution in $K$. The convex hull of the points $x_{1},\ldots,x_{n}$ is a random polytope $K_{n}$ in $K$.

Let $H_{d-1}^{\mathcal{M}^{d}}(x)$ denote the general Gauss–Kronecker curvature at a boundary point $x\in\operatorname{\partial}K$. The boundary of a convex body may not be twice differentiable, so its classical Gauss–Kronecker curvature may not exist at every point. A generalized notion of second-order differentiability can be introduced such that the boundary is differentiable in this sense at almost every point with respect to the surface area measure (cf. Alexandrov’s theorem). Then, a generalized Gauss–Kronecker curvature can be defined at these points that coincides with the classical notion everywhere where the boundary is twice differentiable. The symbol $H_{d-1}^{\mathcal{M}^{d}}(x)$ refers to this generalized notion of Gauss–Kronecker curvature. For more details, see Schütt and Werner [SW23]*Section 2. The spherical and hyperbolic cases of the following theorem were proved by Besau, Ludwig, and Werner [BLW18]*Theorems 2.2, 3.2, and Corollaries 2.3, 3.3. The Euclidean case is due to Schütt [S94] extending results of Bárány [B92].

Theorem 4**.**

Let $K\in\mathcal{K}(\mathcal{M}^{d})$. If $K_{n}$ is the convex hull of $n$ random points chosen uniformly in $K$, then

[TABLE]

where integration on $\partial K$ is with respect to the $(d-1)$-dimensional Hausdorff measure, and $\gamma_{d}$ is an explicitly known constant that depends only on $d$.

We say that a ball of radius $r>0$ rolls freely in $K$ ($K$ slides freely in a ball of radius $R>0$, resp.) if for any $x\in\operatorname{\partial}K$ there exists a ball of radius $r$ ($R$, resp.) containing $x$ on its boundary and contained in $K$ (containing $K$, resp.). If $K$ has a rolling ball and slides freely in a ball at the same time, then $\operatorname{\partial}K$ is $C^{1}$ and strictly convex. However, $\operatorname{\partial}K$ is not necessarily $C^{2}$. A simple example can be constructed in the plane by joining two circular arcs of different radii that share a tangent at their common endpoints, and the other endpoints are connected by a suitable $C^{2}_{+}$ arc. One of our main results is the following theorem.

Theorem 5**.**

Let $K\in\mathcal{K}(\mathcal{M}^{d})$, and assume that $K$ has a rolling ball and slides freely in a ball. Then

[TABLE]

The variance upper bound on the volume implies a strong law of large numbers that can be proved by standard arguments (see, for instance, [BFV10], [R03]).

Corollary 1**.**

Under the same assumptions as in Theorem 5, it holds with probability one that

[TABLE]

The number of vertices is not a monotone function of $n$, but using the fact that it may be increased by at most one when another point is added, for $d\geq 4$, similarly to [R03], a strong law of large numbers can be proved for $f_{0}$ as well. In $\mathbb{R}^{2}$, Bárány and Steiger [BS13] proved variance upper bounds and strong laws of large numbers for the area and the number of vertices of a random polygon with no smoothness condition on $K$. However, the method they used for the upper bounds can not be extended to higher dimensions.

Corollary 2**.**

Under the same assumptions as in Theorem 5, for $d\geq 4$, it holds with probability one that

[TABLE]

Since Besau and Thäle [BT20] proved in $\mathbb{S}^{d}$ and $\mathbb{H}^{d}$ that if $K$ has $C_{+}^{2}$ boundary, then $\operatorname{Var}\operatorname{Vol}_{\mathcal{M}^{d}}(K_{n})\gg n^{-\frac{d+3}{d+1}}$, Theorem 5 yields the following.

Theorem 6**.**

Let $K\in\mathcal{K}(\mathcal{M}^{d})$ with $C_{+}^{2}$ boundary. Then

[TABLE]

For $\mathcal{M}^{d}=\mathbb{R}^{d}$, both the lower and the upper bounds are due to Reitzner [R03, R05].

Our proof of Theorem 5 uses results of a weighted model described in Subsection 2.2, and thus it is indirect.

Circumscribed model

The probability model discussed in this section considers circumscribed spherical polytopes containing a convex body $K$. This model is naturally connected to the inscribed ones discussed via spherical polarity. The role of random points inside of $K$ is replaced by random closed hemispheres containing $K$, and the intersection of $n$ such hemispheres is a random polytope containing $K$. This model was studied, for example, by Besau, Ludwig and Werner [BLW18].

We note that the hyperbolic model is not considered here, as we are not aware of the expectations of the corresponding random variables.

Let $\mathcal{H}$ denote the space of closed hemispheres in $\mathbb{S}^{d}$. Each point $x\in\mathbb{S}^{d}$ is the pole of a unique closed hemisphere $H^{-}(x)=\{y\in\mathbb{S}^{d}:\langle x,y\rangle\leq 0\}$. Thus, $\mathcal{H}$ inherits a natural topology from $\mathbb{S}^{d}$. We define the measure of a Borel set $A\subset\mathcal{H}$ as

[TABLE]

where $\omega_{d+1}$ is the surface volume of $\mathbb{S}^{d}$ and integration is with respect to spherical Lebesgue measure, and $\mathbf{1}(\cdot)$ denotes the indicator function of an event.

Let $\mathcal{H}_{K}=\{H^{-}\in\mathcal{H}:K\subset H^{-}\}$. Choose $n$ i.i.d. random hemispheres containing $K$ according to the uniform distribution with the probability measure $\mu_{K}=\mu/\mu(\mathcal{H}_{K})$. The intersection of these hemispheres is a random polytope containing $K$, denoted by $K^{(n)}$.

The spherical polar $K^{\ast}$ of a convex body $K$ is defined as $K^{\ast}=\bigcap_{x\in K}H^{-}(x).$ Since polarity reverses set inclusion, a hemisphere $H^{-}(x)$ contains $K$ if and only if $x\in K^{\ast}$. The polar body $(K^{(n)})^{\ast}$ is a polytope contained in $K^{\ast}$, and it is the convex hull of those $n$ i.i.d. random points in $K^{\ast}$ that are the poles of the random hemispheres containing $K$. This provides a direct connection between the inscribed and circumscribed models.

Based on this connection, Besau, Ludwig, and Werner [BLW18] proved asymptotic formulas for the expectation of the spherical mean width and the number of facets $f_{d-1}$ of $K^{(n)}$. The spherical mean width $U_{1}(K)$ is defined as

[TABLE]

where $G(d+1,d)$ denotes the Grassmannian of the $d$-dimensional linear subspaces of $\mathbb{R}^{d+1}$, $\nu$ is the unique rotation invariant probability measure on $G(d+1,d)$, and $\chi$ denotes the Euler characteristic.

Theorem 7** ([BLW18], Corollary 2.6).**

Let $K\in\mathcal{K}(\mathbb{S}^{d})$. If $K^{(n)}$ is the intersection of $n$ random hemispheres containing $K$ and chosen uniformly according to $\mu_{K}$, then

[TABLE]

Our variance upper bounds in Theorem 5 yield the following upper bounds on the variances of $U_{1}(K^{(n)})$ and $f_{d-1}(K^{(n)})$ due to spherical polarity.

Corollary 3**.**

Let $K\in\mathcal{K}(\mathbb{S}^{d})$ that has a rolling ball and which slides freely in a ball. Then

[TABLE]

2. Geometric tools

2.1. The economic cap covering theorem

Let $K\in\mathcal{K}(\mathcal{M}^{d})$ and let $H^{-}$ be a closed half-space if $\mathcal{M}^{d}=\mathbb{R}^{d}$ or $\mathbb{H}^{d}$, and a closed hemisphere if $\mathcal{M}^{d}=\mathbb{S}^{d}$. Let $H^{+}$ denote the other closed half-space (hemisphere) determined by $H^{-}$. For $v>0$, we define the convex floating body of $K$ as

[TABLE]

Euclidean convex floating bodies were defined by Bárány and Larman [BL88], and independently by Schütt and Werner [SW04]. The spherical convex floating body was introduced by Besau and Werner [BW16], and hyperbolic convex floating bodies were defined by Besau and Werner [BW18].

The closure of the complement of $K[v]$ in $K$ is called the wet part of $K$ with parameter $v$, and it is denoted by $K(v)$.

In Sections 3, 5, and 8, we are going to use the economic cap covering theorem in $\mathbb{R}^{d}$, that was proved by Bárány and Larman [BL88] and Bárány [Bar89].

Theorem 8** ([BL88], [Bar89]).**

Assume that $K\subset\mathbb{R}^{d}$ is a convex body with $\operatorname{Vol}_{d}(K)=1$ and $0<v<v_{0}=(2d)^{-2d}$. Then there exist caps $C_{1},\ldots,C_{m}$ and pairwise disjoint convex sets $C_{1}^{\prime},\ldots,C_{m}^{\prime}$ such that $C_{i}^{\prime}\subset C_{i}$ for each $i$, and

1. (i)

$\bigcup_{1}^{m}C_{i}^{\prime}\subset K(v)\subset\bigcup_{1}^{m}C_{i}$, 2. (ii)

$\operatorname{Vol}_{d}(C_{i}^{\prime})\gg v$* and $\operatorname{Vol}_{d}(C_{i})\ll v$ for each $i$,* 3. (iii)

for each cap $C$ with $C\cap K[v]=\emptyset$ there is a $C_{i}$ containing C.

2.2. Weighted non-uniform models in $\mathbb{R}^{d}$

Let $K\in\mathcal{K}(\mathbb{R}^{d})$ and let $\varrho:K\to(0,\infty)$ be a probability density function that is continuous in a neighbourhood of the boundary of $K$ (in $K$). Then $\mathbb{P}_{\varrho}(A):=\int_{A}\varrho(x)\,\mathrm{d}x$ for any measurable set $A\subset K$. We denote the expectation and variance with respect to $\mathbb{P}_{\varrho}$ by $\mathbb{E}_{\varrho}$ and ${\rm Var}\,_{\varrho}$, respectively. Let $x_{1},\ldots,x_{n}$ be i.i.d. random points from $K$ distributed according to $\mathbb{P}_{\varrho}$, and let $K_{(n)}$ be their convex hull, a random polytope in $K$.

Furthermore, let $\lambda:K\to(0,\infty)$ be a weight function that is integrable in $K$ and continuous in a neighborhood of $\operatorname{\partial}K$ and let $V_{\lambda}(A)=\int_{A}\lambda(x)\,\mathrm{d}x$ for all measurable subsets $A\subset K$ . For $\varrho\equiv 1/\operatorname{Vol}_{d}(K)$ and $\lambda\equiv 1$, we obtain the uniform model.

In this model, with no smoothness condition on $K$, Böröczky, Fodor and Hug [BFH10] studied the asymptotic behavior of the expectation of the weighted volume of missed part of $K$ and the number of vertices $f_{0}(K_{(n)})$ of the random polytope $K_{(n)}$, and proved that

[TABLE]

Assuming that $\operatorname{\partial}K$ is $C^{2}_{+}$, Besau and Thäle [BT20] proved the following asymptotic variance lower bound and a central limit theorem:

[TABLE]

as $n\to\infty$, where $G$ is a standard normal random variable. The lower bounds in [BT20] for $\mathbb{S}^{d}$ and $\mathbb{H}^{d}$ were deduced from (1) by choosing particular weight functions that come from the gnomonic projections.

Under weaker smoothness assumptions, Bakó-Szabó and Fodor [BSF24] proved matching asymptotic upper bounds for the variance of the weighted volume and the number of vertices.

Theorem 9** ([BSF24], Theorem 1.1).**

For a convex body $K\subset\mathbb{R}^{d}$ that has a rolling ball and which slides freely in a ball, it holds that

[TABLE]

where the implied constants depend only on $K,\varrho,\lambda$ and the dimension $d$.

We use Theorem 9 in our proof of Theorem 5.

2.3. Gnomonic projection

The gnomonic projection maps a $d$-dimensional open hemisphere from the origin radially to a tangent hyperplane. We refer to the point of tangency as the center of the projection. We may assume that the convex body $K\subset\mathbb{S}^{d}$ is contained in the (upper) open hemisphere $\mathbb{S}^{d}_{+}=\{x\in\mathbb{S}^{d}:\langle x,e_{d+1}\rangle>0\}$, and the center of projection is $e_{d+1}$. Then the gnomonic projection $g:\mathbb{S}^{d}_{+}\to\mathbb{R}^{d}$ is

[TABLE]

and the hyperplane $\{x\in\mathbb{R}^{d+1}:\langle x,e_{d+1}\rangle=0\}$ is identified with $\mathbb{R}^{d}$. The map $g$ is bijective and $C^{\infty}$. Geodesic arcs of $\mathbb{S}^{d}_{+}$ are mapped into straight line segments in $\mathbb{R}^{d}$, thus the image of a spherical convex body is a convex body in the Euclidean sense. The gnomonic image of a set $X$ is often denoted by $\overline{X}=g(X)$.

The gnomonic projection can be defined in the hyperbolic model similarly: $h:\mathbb{H}^{d}\to\mathbb{R}^{d}$ maps the points of the hyperboloid centrally to the tangent hyperplane $H=\{x\in\mathbb{R}^{d+1}:\langle x,e_{d+1}\rangle=1\}$, then identify $H$ with $\mathbb{R}^{d}$, i.e. identify

[TABLE]

As $\|h(x)\|<1$ for all $x\in\mathbb{H}^{d}$, the image $g(\mathbb{H}^{d})$ is the open unit ball of $\mathbb{R}^{d}$. The map $h$ is $C^{\infty}$, geodesic arcs are mapped into straight line segments, and it is bijective between $\mathbb{H}^{d}$ and $\operatorname{int}B^{d}$. The image of a hyperbolic convex body is a convex body in $\mathbb{R}^{d}$.

3. Proof of the upper bound in Theorem 1

For a measurable set $A\subset\mathbb{R}^{d}$, the probability that a beta-distributed random point falls in $A$ is called the probability content (or beta-content) of $A$, denoted by $\mathbb{P}_{\beta}(A)$. A crucial part of the proof is to determine the order of magnitude of the probability content of a cap $C$ of height $t$.

Lemma 1**.**

Let $\varepsilon>0$ and $0<t\leq 1-\varepsilon$. Then

[TABLE]

Proof.

We define the following functions: the one-dimensional cumulative distribution function $F_{1,\beta}$, the beta function $B(a,b)$ and the incomplete beta function $B(z;a,b)$.

[TABLE]

Then, after the substitution $y=1-x^{2}$, the following hold:

[TABLE]

According to [KTT19]*Lemma 4.5.,

[TABLE]

We apply the following series expansion from [AS64]*Equation 26.5.4 on incomplete beta functions with $0<z<1$.

[TABLE]

Therefore,

[TABLE]

Notice that for fixed $b$, the beta function $B(a,b)$ is monotonically decreasing in $a$. Hence,

[TABLE]

where the right-hand side is bounded from above by $\frac{1}{\varepsilon^{2}}-1$ since $t\leq 1-\varepsilon$.

Thus, we obtain the following inequalities on $\mathbb{P}_{\beta}(C(t))$.

[TABLE]

∎

Later in this paper, we are going to apply the economic cap covering theorem, which requires its parameter $v$ to be at most $\kappa_{d}(2d)^{-2d}$. Thus, the height $t$ of the caps considered will be bounded by a constant depending on $d$ (and decreasing as $d$ increases). We may set $\varepsilon$ to a fixed value; for example, $\varepsilon=1/2$ will be suitable, as we only work with small $t$.

For the volume of a spherical cap $C(t)$ of height $t$, we have $\operatorname{Vol}_{d}(C(t))\approx t^{\frac{d+1}{2}}$. Hence, the volume of a cap can be converted to beta-content as follows. There exist positive constants $\gamma_{1}$ and $\gamma_{2}$, such that if $\operatorname{Vol}_{d}(C(t))=v$, then

[TABLE]

We recall a technical lemma from [BFV10]. For given $z\in\mathbb{S}^{d-1}$ and $A\in G(d,s)$, their angle $\angle(z,A)$ is defined as the minimal angle $\angle(z,x)$ over all $x\in A$.

Lemma 2** ([BFV10] Lemma 1, p. 609.).**

Let $z\in\mathbb{S}^{d-1}$ and (small) $\alpha>0$ be fixed, then $\nu_{s}\{A\in G(d,s)\,|\,\angle(z,A)\leq\alpha\}\approx\alpha^{d-s}$.

Let $T_{n}$ denote the event that the floating body $\widetilde{B}=B^{d}[(c\log n/n)^{\frac{d+1}{d+1+2\beta}}]$, which is a concentric ball, is contained in $K_{n}^{\beta}$ for a suitable constant $c$ that will be specified later. There is a constant $\delta$ such that the probability of $T_{n}^{c}$, i.e., the complement of $T_{n}$, is at most $n^{-\delta c}$. This can be seen as follows.

Suppose that $\widetilde{B}$ has a point $x$ in $B^{d}\setminus K_{n}^{\beta}$. Let $C$ be the cap cut off by the half-space $H^{+}$ for which $x$ is the center of the $(d-1)$-dimensional ball $B^{d}\cap H$. If $K_{n}^{\beta}\subset H^{-}$, then $C$ does not contain any of the random points. If this is not the case, we show that there is an empty part of the cap $C$ with sufficiently large beta-content. We may assume that $x$ is in the direction of $\textbf{e}_{d}$. Thus, those coordinate hyperplanes of the standard Cartesian coordinate system that contain $x$ cut $C$ into $2^{d-1}$ identical pieces, and $B^{d}\setminus C$ into another $2^{d-1}$ larger pieces. If all of these parts contained a random point, then $x$ would be in their convex hull. Thus, there must be at least one empty coordinate corner, and its beta-content is at least $2^{-(d-1)}$ times that of the cap $C$, by symmetry.. However, the definition of floating body yields $\operatorname{Vol}_{d}(C)\geq(c\log n/n)^{\frac{d+1}{d+1+2\beta}}$, and $\mathbb{P}_{\beta}(C)\geq\gamma_{1}c\log n/n$. Then the probability of $T_{n}^{c}$ is at most $\left(1-\frac{\gamma_{1}}{2^{d-1}}c\log n/n\right)^{n}\leq n^{-\delta c}$ for some $\delta>0$, which depends only on $d$ and $\gamma_{1}$. In the uniform case, more precise asymptotics of probabilities are proven in [BD97].

Finally, we choose $c$ to be sufficiently large, so the conditional expectation on the event $T_{n}^{c}$ can be omitted.

We estimate the variance of the intrinsic volumes $V_{s}(K_{n}^{\beta})$ from above using the Efron-Stein jackknife inequality [ES81] and Kubota’s formula

[TABLE]

where $K|A$ denotes the orthogonal projection of $K$ onto $A$ and $C_{d,s}$ is a constant depending only on $d$ and $s$.

[TABLE]

Let $X_{m}$ denote the collection of the first $m$ random points $x_{1},\ldots,x_{m}$, for $m\in\{1,\ldots,n+1\}$. For an index set $I=\{i_{1},\ldots,i_{s}\}\subset\{1,\ldots,n\}$, let $F_{I}$ denote the convex hull of $x_{i_{1}},\ldots,x_{i_{s}}$, which is an $(s-1)$-dimensional simplex with probability $1$. Note that $(K_{n+1}^{\beta}|A)\setminus(K_{n}^{\beta}|A)$ is either empty or a union of simplices determined by $x_{n+1}|A$ and the facets of $K_{n}^{\beta}|A$ that can be seen from $x_{n+1}|A$. We say that a facet $F$ of $K_{n}^{\beta}|A$ is visible from a point $x\in A\setminus(K_{n}^{\beta}|A)$, if the open segment between $x$ and a point in the relative interior of $F$ is disjoint from $K_{n}^{\beta}|A$. Let $U_{A}$ (or $U_{B}$) denote the event that the ($s-1$)-dimensional facet $F_{I}|A$ (or $F_{J}|B$, resp.), of $K_{n}^{\beta}|A$ is visible from $x_{n+1}|A$.

The right-hand side of (3) can be estimated as

[TABLE]

where the summation goes over all $s$-tuples $I$ and $J$.

Similarly to [BFV10], we denote the smaller $s$- and $d$-dimensional caps cut off by the affine hull of $F_{I}|A$ or $A^{\perp}+{\rm aff}\,F_{I}$, resp. from $B^{d}$ by $C_{s}(I,A)$ and $C_{d}(I,A)$ respectively. For their volumes we use $V_{s}(I,A)=\operatorname{Vol}_{s}(C_{s}(I,A))$ and $V_{d}(I,A)=\operatorname{Vol}_{d}(C_{d}(I,A))$, moreover, for the probability content we use $\mathbb{P}_{\beta}(I,A)=\mathbb{P}_{\beta}(C_{d}(I,A))$.

The volumes of the simplices $[F_{I},x_{n+1}]|A$ and $[F_{J},x_{n+1}]|B$ can be estimated from above by the volumes of the caps $C_{s}(I,A)$ and $C_{s}(J,B)$, respectively.

[TABLE]

As the summation goes over all $s$-tuples $I$ and $J$, $I$ and $J$ may intersect. Let the number of common elements of $I$ and $J$ be $k$.

Without loss of generality, we may choose $I=\{1,\dots,d\}$ and $J=\{s-k+1,\dots 2s-k\}$, as the corresponding terms in (5) are independent of the choice of $i_{1},\ldots,i_{d}$ and $j_{1},\ldots,j_{d}$. Thus,

[TABLE]

By symmetry, we may assume the event $L$ that $V_{s}(I,A)\geq V_{s}(J,B)$. Therefore

[TABLE]

Since $C_{d}(I,A)$ and $C_{d}(J,B)$ have at least $x_{n+1}$ in common, replacing $\mathbf{1}(U_{B})$ to $\mathbf{1}(N):=\mathbf{1}(C_{d}(I,A)\cap C_{d}(J,B)\neq\emptyset)$ in the integrand does not decrease the integral. We estimate the terms in the sum separately for each $k=0,\ldots,d$, denoted by $\Sigma_{k}$.

[TABLE]

If $U_{A}$ holds, then all the points $x_{2s-k+1},\ldots,x_{n}$ must be contained in $B^{d}\setminus C(I,A)$ and $x_{n+1}$ has to be contained in $C(I,A)$. Then we can integrate with respect to $x_{2s-k+1},\ldots,x_{n},x_{n+1}$, and the condition $T_{n}$ can be replaced by the condition $W_{n}=\{\mathbb{P}_{\beta}(I,A)\leq\gamma_{2}c\log n/n\}$, where the notation $\{\mathcal{S}\}$ for a probability event denotes the set of those outcomes of the random experiment for which the statement $\mathcal{S}$ holds.

[TABLE]

We can use the geometric estimations in [BFV10]*p. 615 that do not depend on the probability distribution; i.e., the conditions $N$ and $L$ imply that the cap $C_{d}(I,A)$ can be blown up from its center by a constant in order to contain $C_{d}(J,B)$ as well; furthermore, the angle between the subspace $B$ and the center $z$ of the cap $C_{d}(I,A)$ has to be at most $b_{d}V_{d}(I,A)^{1/(d+1)}$ for a constant $b_{d}$ depending only on $d$. Therefore, after integrating with respect to the variables $x_{i}$, $i\in J\setminus I$, we obtain that

[TABLE]

Now we are going to estimate the inner integral for a fixed $A\in G(d,s)$ using the economic cap covering theorem. Due to the condition $W_{n}$, every cap $C_{d}(I,A)$ has beta-content at most $\gamma_{2}c\log n/n$.

We follow the standard argument: for each positive integer $h$ for which $2^{-h}\leq(c\log n)/n$, let $\mathcal{C}_{h}$ be a collection of caps $\{C_{1},\ldots,C_{m(h)}\}$ forming the economic cap covering of the wet part of $B^{d}|A$ with $v=(2^{-h})^{\frac{s+1}{d+1+2\beta}}$ (we assume that $n$ is sufficiently large). For the $d$-dimensional caps $C_{i}^{\prime}$ in $B^{d}$, whose projections to $A$ are $C_{i}$, we have that $\operatorname{Vol}_{d}(C_{i}^{\prime})\ll(2^{-h})^{\frac{d+1}{d+1+2\beta}}$. Consider an arbitrary $(x_{1},\ldots,x_{s})$ with the corresponding $C_{d}(I,A)$ having beta-content at most $\gamma_{2}c\log n/n$, and associate with $(x_{1},\ldots,x_{s})$ the maximal $h$ such that for some $C_{i}\in\mathcal{C}_{h}$, $C_{s}(I,A)\subset C_{i}$. Such an $h$ clearly exists. Then

[TABLE]

and

[TABLE]

On the other hand, by the maximality of $h$,

[TABLE]

The integrand in (8) can be estimated by

[TABLE]

and, after integrating each $(x_{1},\ldots,x_{s})$ on its associated $C_{i}^{\prime}$, the inner integral is bounded by

[TABLE]

The final step in the proof is to calculate the number of elements of $\mathcal{C}_{h}$. The volume of the wet part of $B^{s}$ (a spherical shell) with parameter $2^{-h\frac{s+1}{d+1+2\beta}}$ is $\operatorname{Vol}_{s}(B^{s}(2^{-h\frac{s+1}{d+1+2\beta}}))\approx 2^{\frac{-2h}{d+1+2\beta}}$. (The $\approx$ notation makes sense, since $h\to\infty$ as $n\to\infty$). Therefore,

[TABLE]

Assembling the pieces estimating the two inner integrals of (8), together with the condition $\angle(z,B)\leq 2b_{d}V_{d}(I,A)^{1/(d+1)}$ and applying Lemma 2 and (9), we obtain with $h_{0}=\left\lfloor c\log n/n\right\rfloor$,

[TABLE]

Calculating the sum of the series is very similar to [BFV10]*p. 617, with slightly different exponents. The order of magnitude of the sum in (10) is $n^{-2s+k-1}n^{-\frac{d+3}{d+1+2\beta}}$, therefore

[TABLE]

This holds for all $k=0,\ldots,d$, thus the sum of them is of the same order of magnitude, which proves the theorem.

4. Proof of the lower bound in Theorem 1

The proof of the lower bound essentially follows a method presented by Reitzner [R05]: one defines small caps independently of each other and provides a lower bound on the variance in each cap. This method has been used in various settings; see, for instance, [BFV10], [BT20], and [FGV22]. As the geometry of the intrinsic volumes and $B^{d}$ is the same as in [BFV10]*Section 4, we only highlight the differences that are due to in the probability model.

For a very small $t>0$, we take a cap $C(x,t)$ in $B^{d}$ of height $t$, whose center is $x\in\mathbb{S}^{d-1}$. Let the hyperplane that cuts off $C(x,t)$ be $H(x,t)$. Then we consider a regular $(d-1)$-simplex inscribed into $B(x,t)=B^{d}\cap H(x,t)$, call its vertices $w_{1},w_{2},\ldots,w_{d}$. Furthermore, let $w_{0}=x$. Then $\Delta=[w_{0},w_{1},\ldots,w_{d}]$ is a $d$-dimensional simplex inscribed in $C(x,t)$. We define the smaller simplices as

[TABLE]

for $j=0,1,\ldots,d$. Each $\Delta_{j}$ is a homothetic copy of $\Delta$ obtained by a homothety with center at $w_{j}$ and factor $1/(4d)$, see Figure 1. We need to estimate the beta-content of $\Delta_{j}$.

Lemma 3**.**

For $j=0,\ldots,d$,

[TABLE]

Proof.

On the one hand, each $\Delta_{j}$ is contained in a cap which has a probability content of order $t^{\frac{d+1+2\beta}{2}}$, hence the upper bound. On the other hand, we prove the lower bound for the case $j=0$, and indicate the modifications required for the other cases.

We are going to estimate the integral $\int_{\Delta_{0}}(1-\|x\|^{2})^{\beta}\,\mathrm{d}x$. We may assume that the apex $w_{0}$ of the simplex is $(0,\ldots,0,1)$, and the base is a regular $(d-1)$-dimensional simplex inscribed in a $(d-1)$-dimensional ball of radius $\sqrt{\frac{t}{4d}(2-\frac{t}{4d})}$ centered at $(0,\ldots,0,1-\frac{t}{4d})$ in the hyperplane $x_{d}=1-\frac{t}{4d}$. We denote the points in $\Delta_{0}$ by $x=(x^{\prime},y):=(x_{1},\ldots,x_{d-1},y)$, i.e. $x^{\prime}\in\mathbb{R}^{d-1}$, to emphasize the role of the $d$-th coordinate. Then $\|x\|^{2}=y^{2}+\|x^{\prime}\|^{2}$, and

[TABLE]

where $\Delta_{d-1}\Big(\sqrt{\frac{t}{4d}(2-\frac{t}{4d})}\Big)$ is the regular $(d-1)$-dimensional simplex inscribed in the $(d-1)$-dimensional ball of radius $\sqrt{\frac{t}{4d}(2-\frac{t}{4d})}$; and $\delta=\delta(y)=\frac{4d}{t}(1-y)$ is the linear factor of homothety in $y$.

We apply the substitution $y=1-u$, then

[TABLE]

Then we use the fact that the height of $\Delta_{j}$ is of order $t$, and the linear size of the base is proportional to $\sqrt{t}$, by substituting $u=\frac{vt}{4d}$ and $x^{\prime}=w\sqrt{\frac{t}{4d}}$. Then

[TABLE]

Since $t\to 0$, we may assume that $t\leq 4d\varepsilon_{1}$, for some constant $\varepsilon_{1}$. Then $\Delta_{d-1}\Big(\sqrt{2-\frac{t}{4d}}\Big)\supseteq\Delta_{d-1}\big(\sqrt{2-\varepsilon_{1}}\big)$. Since the integrand, inherited from the definition of the beta distribution, is non-negative, restricting the integration to $v\cdot\Delta_{d-1}\big(\sqrt{2-\varepsilon_{1}}\big)$ does not increase the integral. Now we divide the proof into two parts based on the sign of $\beta$.

First, let $\beta<0$. Then, estimating $\Big(2v-\frac{v^{2}t}{4d}-\|w\|^{2}\Big)$ from above gives a lower bound for the integrand. Therefore,

[TABLE]

The integral in (14) is now independent of $t$, hence the right-hand side is a positive constant times $t^{\frac{d+1+2\beta}{2}}$.

Now, let $\beta\geq 0$. In this case, we need to estimate the integrand in (4) from below. Thus we obtain

[TABLE]

where the integral is again independent of $t$. Since $\|w\|^{2}\leq v^{2}(2-\varepsilon_{1})$, the integrand is bounded from below by $2(v-v^{2})$, which is positive for $v\in(0,1)$. Thus, the right-hand side is a positive constant times $t^{\frac{d+1+2\beta}{2}}$.

The cases $j=1,\ldots,d$ are similar, relying on the same scaling argument; however, the integration domain in (11) needs to be modified in the following way. The $(d-1)$-dimensional ball of radius $\sqrt{\frac{t}{4d}(2-\frac{t}{4d})}$ circumscribed of the regular $(d-1)$-simplex is not centered, but translated such that one vertex of the simplex lies on the boundary of the $d$-dimensional unit ball. Furthermore, the homothety, which is linear in $y$, also changes the center of $\Delta_{d-1}$ together with its size. Moreover, $y$ varies between $1-t$ and $1-t+\frac{t}{4d}$, and in (12), we substitute $u=1-t+\frac{t}{4d}-y$.

∎

We choose a point $z_{j}$ in each $\Delta_{j}(x,t)$, fix $x$, $t$ and $z_{j}\in\Delta_{j}(x,t)$ for $j=1,\ldots,d$, and write $F=[z_{1},\ldots,z_{d}]$. Similarly to [BFV10], we define the function $\hat{V}_{s}:\Delta_{0}(x,t)\to\mathbb{R}$ as follows

[TABLE]

where $\Sigma_{2}(x,t)=\mathbb{S}^{d-1}\cap\left(x+2d\sqrt{t}B^{d}\right).$

Lemma 4**.**

If $Z$ is a random point chosen according to the normalized beta-distribution on $\Delta_{0}(x,t)$, then

[TABLE]

For the proof, we refer to [BFV10]*Lemma 2, as it is mainly geometric and the distribution does not make a difference.

We set

[TABLE]

then, by Lemma 1, $\mathbb{P}_{\beta}(C(x,t_{n}))\approx 1/n$ for all $x\in\mathbb{S}^{d-1}$. Let $y_{1},\ldots,y_{m}\in\mathbb{S}^{d-1}$ be a maximal set of points such that $d(y_{i},y_{j})\geq 2\sqrt{\gamma}\sqrt{t_{n}}$ for $i,j\in\{1,\ldots,m\}$ and for a constant $\gamma$ specified as follows. Let $\gamma$ be so large that the caps $C(y_{j},\gamma t_{n})$ ($j=1,\ldots,m$) are pairwise disjoint. Then

[TABLE]

Inscribe the simplex $\Delta(y_{j},t_{n})$ in the cap $C(y_{j},t_{n})$ for each $j\in\{1,\ldots,m\}$, and construct the small simplices $\Delta_{i}(y_{j},t_{n})$. Let $A_{j}$ denote the event that each $\Delta_{i}(y_{j},t_{n})$, $i=0,\ldots,d$, contains exactly one random point out of $x_{1},\ldots,x_{n}$, and no other random points are contained in $C(y_{j},\gamma t_{n})$. By Lemmas 1 and 3, we have

[TABLE]

Therefore, for $j=1,\ldots,m$, it holds that

[TABLE]

Similarly to [BFV10], we obtain

[TABLE]

5. Proof of Theorem 2

The proof essentially follows the method of Thäle, Turchi and Wespi [TTW18]; Besau and Thäle [BT20] adapted it to non-Euclidean geometries, and Fodor and Papvári [FP24] used a similar argument for a spindle convex model in $\mathbb{R}^{2}$. We apply the normal approximation bounds that were used by [Cha08] and [LRP17].

Let $f:\bigcup_{k=1}^{n}(B^{d})^{k}\to\mathbb{R}$ be a measurable and symmetric function that acts on point configurations of at most $n\in\mathbb{N}$ points in $B^{d}$. For $x=(x_{1},\ldots,x_{n})\in(B^{d})^{n}$, we define the first- and second-order difference operators applied to $f(x)=f(x_{1},\ldots,x_{n})$ as

[TABLE]

respectively, where $x^{i}$ denotes the $(n-1)$-dimensional vector one gets by removing the $i$-th coordinate of $x$, and similarly, $x^{i,j}$ arises by removing the $i$-th and $j$-th coordinates of $x$.

For a random vector $X=(x_{1},\ldots,x_{n})$ of elements of $B^{d}$, we introduce the random copies $X^{\prime}$ and $X^{\prime\prime}$ of $X$. Let the random vector $Z=(z_{1},\ldots,z_{n})$ be a recombination of $\{X,X^{\prime},X^{\prime\prime}\}$, i.e., for all $i\in\{1,\ldots,n\}$, $z_{i}\in\{x_{i},x^{\prime}_{i},x^{\prime\prime}_{i}\}$ .

We need the following quantities:

[TABLE]

where the suprema in the first two definitions are taken over all $4$- and $3$-tuples of vectors $(Y,Y^{\prime},Z,Z^{\prime})$ and $(Y,Z,Z^{\prime})$, respectively, that are recombinations of $\{X,X^{\prime},X^{\prime\prime}\}$.

Let $W=f(x_{1},\ldots,x_{n})$ and assume that $W$ is centered, with finite positive second moment. Let $G$ denote a standard Gaussian random variable. Then, by [LRP17],

[TABLE]

In our case, for a fixed $\beta>-1$, let $x_{1},\ldots,x_{n}$ be i.i.d. random points according to the beta-distribution, and for a fixed $s\in\{1,\ldots,d\}$, let $W=V_{s}(K_{n}^{\beta})-\mathbb{E}V_{s}(K_{n}^{\beta})$. Then $\mathbb{E}W=0$, and by Theorem 1, the variance is finite.

As in Section 3, let $T_{n}$ denote the event that the floating body $\widetilde{B}=B^{d}[(c\log n/n)^{\frac{d+1}{d+1+2\beta}}]$ is contained in $K_{n}^{\beta}$ for a suitable constant $c$. Then there is a constant $\delta$ such that the probability of $T_{n}^{c}$ is at most $n^{-\delta c}$. Similarly, let $\tau_{n}$ denote the event that $\widetilde{B}$ is contained in $\bigcap_{W\in\{Y,Y^{\prime},Z,Z^{\prime}\}}[W_{4},\ldots,W_{n}]$. The probability of $\tau_{n}^{c}$ is smaller than $n^{-\delta^{\prime}c}$ for a constant $\delta^{\prime}$.

The main difference between our model and the model studied in [TTW18] is due to the probability distribution of the random points, and not in the geometry.

We estimate the first-order difference operator using Kubota’s formula (2):

[TABLE]

where $K_{n-1}^{\beta}$ now denotes $[x_{2},\ldots,x_{n}]$. First, we need to calculate the probability of the event that, assuming $T_{n}$, $x_{1}$ is not in $K_{n-1}^{\beta}$, because otherwise the integrand in (16) is zero. Assuming $T_{n}$, we have the following.

[TABLE]

Therefore, we need to estimate the beta-content of the wet part of $B^{d}$ with parameter $v=(c\log n/n)^{\frac{d+1}{d+1+2\beta}}$. Recall that the beta content of a cap $C$ with volume at most $v$ is at most $\gamma_{2}c\log n/n$.

Theorem 8 (economic cap covering theorem) states that the wet part $B^{d}\setminus\widetilde{B}$ can be covered with caps of volume $v$, and the number of caps needed is of order at most $v^{-\frac{d-1}{d+1}}$. We can estimate the probability content of the wet part from above by taking the probability content of all the caps used in the covering. Thus,

[TABLE]

Now, let $z\in\mathbb{S}^{d-1}$ be the closest point to $x_{1}$ on the boundary of $B^{d}$, and define its visibility region as

[TABLE]

Then for the diameter of the visibility region, it holds that $\operatorname{diam}\operatorname{Vis}_{z}(n)\ll(\log n/n)^{\frac{1}{d+1+2\beta}}$ (cf. [Vu05]*Lemma 6.2). Similarly to [TTW18], we construct the cap $C$, which is the convex hull of all the points of $\mathbb{S}^{d-1}$, whose distance from $z$ is at most $\operatorname{diam}\operatorname{Vis}_{z}(n)$. For the central angle $\alpha$ of $C$, we have $\alpha\ll\big(\frac{\log n}{n}\big)^{\frac{1}{d+1+2\beta}}$ and the integrand in (16) can only be positive if $\angle(z,A)\ll\alpha$. By Lemma 2, $\nu_{s}\{A\in G(d,s)\,|\,\angle(z,A)\ll\big(\frac{\log n}{n}\big)^{\frac{1}{d+1+2\beta}}\}\ll\big(\frac{\log n}{n}\big)^{\frac{d-s}{d+1+2\beta}}$. Although the projection of $C$ onto a subspace $A\in G(d,s)$ is generally not a cap, a simple trigonometric calculation shows that, assuming $\angle(z,A)\ll\alpha$, $C|A$ is contained in a cap whose height is of the same order as the height of $C$. We can estimate the volume of $C$ and the $s$-dimensional volume of its projection as

[TABLE]

Since $(K_{n}^{\beta}|A)\setminus(K_{n-1}^{\beta}|A)\subset C|A$, (16) can be estimated from above as

[TABLE]

Therefore, we estimate the expectation of $\big(D_{1}V_{s}(K_{n}^{\beta})\big)^{p}$ as follows. In the definition of $\widetilde{B}$, we choose $c$ to be sufficiently large, so the conditional expectation on the event $T_{n}^{c}$ can be omitted. For $p\in\{1,\ldots,4\}$,

[TABLE]

From (5) we also obtain that, assuming $\tau_{n}$,

[TABLE]

for all $i\in\{1,2,3\}$ and $Z^{\ast}\in\{Z,Z^{\prime}\}$. We recall from [TTW18], that assuming $\tau_{n}$, it holds for $j\in\{2,3\}$ and $Y^{\ast}\in\{Y,Y^{\prime}\}$, that

[TABLE]

and

[TABLE]

The previous union is contained in a cap whose diameter is of order $\big(\frac{\log n}{n}\big)^{\frac{1}{d+1+2\beta}}$, so its height is of order $\big(\frac{\log n}{n}\big)^{\frac{2}{d+1+2\beta}}$, and therefore its beta-content can be estimated as

[TABLE]

Analogously, assuming $\tau_{n}$, it holds for $y_{1}=y_{1}^{\prime}$, that

[TABLE]

and, since the case $y_{1}\neq y_{1}^{\prime}$ gives a smaller factor,

[TABLE]

Now, using (19)–(23), we obtain

[TABLE]

The terms in (15) can be estimated from above as

[TABLE]

Thus, for the Wasserstein distance, we obtain

[TABLE]

6. Proof of Theorem 3

First, we sketch the proof of the upper bound. Our argument is similar to that of Reitzner [R05, R05b], and is based on a suitable modification of the proof of the upper bound of Theorem 1 in Section 3.

The variance is again estimated from above using the Efron-Stein inequality.

[TABLE]

where $T_{n}$ is the condition that the floating body $\widetilde{B}$ is contained in $K_{n}^{\beta}$.

Let $F_{n}$ denote the number of facets of $K_{n}^{\beta}$ that can be seen from $x_{n+1}$. If $x_{n+1}\in K_{n}^{\beta}$, then $F_{n}=0$ and $K_{n+1}^{\beta}$ has the same number of $k$-dimensional faces as $K_{n}^{\beta}$. Otherwise, $F_{n}$ is positive. For each $k\in\{0,1,\ldots,d-1\}$, let $f_{k}^{+}$ denote the number of $k$-faces of $K_{n+1}^{\beta}$ that are not contained in $K_{n}^{\beta}$, and let $f_{k}^{-}$ denote the number of $k$-faces of $K_{n}^{\beta}$ that are not faces of $K_{n+1}^{\beta}$. We recall from [R05b], that $|f_{k}(K_{n+1}^{\beta})-f_{k}(K_{n}^{\beta})|\leq f_{k}^{+}+f_{k}^{-}$, furthermore,

[TABLE]

therefore, $\mathbb{E}[(f_{k}(K_{n+1}^{\beta})-f_{k}(K_{n}^{\beta}))^{2}]\ll\mathbb{E}[F_{n}^{2}].$

We recall some notation from Section 3 with $s=d$ (in this case, the projections are the identity map, and therefore, not indicated): $I,J\subset\{1,\ldots,n\}$ are index sets with $d$ elements each, $F_{I}$ and $F_{J}$ denote the convex hulls of $x_{i_{1}},\ldots,x_{i_{d}}$, and $x_{j_{1}},\ldots,x_{j_{d}}$, respectively. Moreover, $U_{A}$ and $U_{B}$ denote the events that $F_{I}$ and $F_{J}$, resp., is visible from $x_{n+1}$. Then (24) can be estimated from above as follows.

[TABLE]

The integral is very similar to (3), the only differences are that now $s=d$, the projections are removed, and the terms $\operatorname{Vol}_{d}([F_{I},x_{n+1}])$ and $\operatorname{Vol}_{d}([F_{J},x_{n+1}])$ are missing. A calculation analogous to the one in the proof of the upper bound in Theorem 6 yields an upper bound of order $n^{\frac{d-1}{d+1+2\beta}}$.

The proof of the lower bound of $\operatorname{Var}f_{k}$ is based on that of the lower bound of the variance of the intrinsic volumes in Section 4, and follows the argument of [R05] as well, suitably modified for the beta-distribution.

Let the caps $C(y_{j},\gamma t_{n})$ be defined as in Section 4, as well as the simplices $\Delta_{i}(y_{j},t_{n})$, for $j=1,\ldots,m$ and $i\in\{0,1,\ldots,d\}$. Let $B_{j}$ denote the event that $\Delta_{0}(y_{j},t_{n})$ contains two random points out of $x_{1},\ldots,x_{n}$, each $\Delta_{i}(y_{j},t_{n})$, $i=1,\ldots,d$ contains one random point, and no other random points are contained in $C(y_{j},\gamma t_{n})$. For the probability of $B_{j}$, $j=1,\ldots,m$, by Lemmas 1 and  3, we have

[TABLE]

Assuming $B_{j}$, let $Y_{j}$ and $Z_{j}$ denote the points in $\Delta_{0}(y_{j},t_{n})$, and let $z_{j,1},\ldots,z_{j,d}$ denote the points in $\Delta_{i}(y_{j},t_{n})$, $i=1,\ldots,d$. For a fixed $j$, the convex hull of $\{Y_{j},Z_{j},z_{j,1},\ldots,z_{j,d}\}$ is either a simplex, or both $Y_{j}$ and $Z_{j}$ are vertices. Both of them arise with positive probability. Thus, $f_{k}([Y_{j},Z_{j},z_{j,1},\ldots,z_{j,d}])$ can take at least two different values with positive probability. Thus, taking the variance for $Y_{j}$ and $Z_{j}$,

[TABLE]

Then, similarly to [R05], we obtain

[TABLE]

7. Proof of Theorem 5

7.1. The spherical case

Assume that a spherical ball of radius $r$ rolls freely in $K$. Let $e_{d+1}$ be the center of the circumsphere of $K$. Then $K$ is contained in the upper hemisphere.

For a boundary point $x\in\operatorname{\partial}K$, denote by $B_{x}$ the ball of radius $r$ for which $x\in\operatorname{\partial}B_{x}$ and $B_{x}\subset K$. Then $g$ maps balls in $\mathbb{S}^{d}_{+}$ to ellipsoids in $\mathbb{R}^{d}$, and those balls, whose center is $e_{d+1}$, are mapped to Euclidean balls in $\mathbb{R}^{d}$. Thus, for each $x\in\operatorname{\partial}K$, $\overline{B}_{x}=g(B_{x})$ is an ellipsoid in $\overline{K}$. Due to compactness, there is a maximal $\kappa_{\max}$ among the principal curvatures of all $\overline{B}_{x}$. Then, a Euclidean ball of radius $1/\kappa_{\max}$ rolls freely in any $\overline{B}_{x}$ (see [Sch14]*Corollary 3.2.13.) and, in turn, it rolls freely in $\overline{K}$, thus $\overline{K}$ also has a rolling ball.

Now, assume that $K$ slides freely in a spherical ball of radius $R$. Then its spherical polar body $K^{\ast}$ has a rolling ball of radius $(\pi/2-R)$ This can be seen in the following way: For a boundary point $x$ on $\operatorname{\partial}K^{\ast}$, there is at least one supporting hypersphere $G^{\ast}$. Its polar $G$ is a point on the boundary of $K$. Since $K$ has a sliding ball, a ball $B_{G}$ of radius $R$ contains $K$, and $G$ is on its boundary. Then $B_{G}^{\ast}$ is a ball of radius $(\pi/2-R)$ contained in $K^{\ast}$ and intersects $G^{\ast}$ at a boundary point $y$ of $K^{\ast}$. If $y\neq x$ then $\operatorname{\partial}K^{\ast}$ contains a great circle segment, thus $\operatorname{\partial}K$ must have a point where it is not smooth. However, this contradicts the assumption that $K$ has a rolling ball, therefore $x=y$, and $K^{\ast}$ also has a rolling ball.

Note that the assumptions on $K$ yield that the rolling ball inside $K^{\ast}$ is in the lower open hemisphere for all $x\in\operatorname{\partial}K^{\ast}$.

Let $\tilde{g}$ be the gnomonic projection whose center is $-e_{d+1}$. Let $(\cdot)^{\circ}$ denote Euclidean polarity in $\mathbb{R}^{d}$. It is known, see Schneider [Sch22]*Lemma 3.2.2., that $g(K)^{\circ}=\tilde{g}(K^{\ast})$. By the above argument, the existence of a rolling ball of $K^{\ast}$ implies that its image, $\tilde{g}(K)^{\circ}$ has a rolling ball too. Hug [H00]*Prop. 1.45. proved that if a convex body $L\subset\mathbb{R}^{d}$ has a rolling ball, then $L^{\circ}$ slides freely in a ball of finite radius. Hence, $g(K)$ slides freely in a ball.

The gnomonic projection maps $\operatorname{Vol}_{\mathbb{S}^{d}}$ into the Lebesgue measure in $\mathbb{R}^{d}$ with the density $\psi(x)=(1+\|x\|^{2})^{-(d+1)/2}$ (see [BW16]*Proposition 4.2), and $\operatorname{Vol}_{\mathbb{S}^{d}}(K_{n})$ has the same distribution as the weighted volume of the convex hull of $n$ i.i.d. random points in $\mathbb{R}^{d}$ (cf. [BT20]*Section 5.1). Thus, we may apply Theorem 9, as the boundary conditions are satisfied for $\overline{K}$.

7.2. The hyperbolic case

Assume that the center of a maximal radius inscribed ball $B_{1}$ in $K$ is the point $e_{d+1}$. Then $h(e_{d+1})=o$ and $h(B_{1})=\overline{B}_{1}\subset\operatorname{int}B^{d}$ in $\mathbb{R}^{d}$.

Suppose that $K$ slides freely in a ball of radius $R$. Let $B_{2}$ denote the smallest ball centered at $e_{d+1}$ that contains the union of all the sliding balls of radius $R$ of $K$. The boundary of $K$ is contained in the spherical shell determined by the concentric balls $B_{1}$ and $B_{2}$, and $\operatorname{\partial}\overline{K}$ is in the (Euclidean) spherical shell determined by $\overline{B}_{1}$ and $\overline{B}_{2}$.

The gnomonic image of a hyperbolic ball contained in $B_{2}$ is an ellipsoid in $\overline{B}_{2}$ (contained in $\operatorname{int}B^{d}$ in $\mathbb{R}^{d}$). It is clear that in the images of such balls the ratio of the ellipsoid’s largest and shortest axes is bounded from above.

For a fixed boundary point $x\in\operatorname{\partial}K$, the image of the sliding ball $B_{x}$ of $K$ at $x$ is an ellipsoid that slides freely in a Euclidean ball of radius $R_{x}$. By compactness, there is a maximum radius $\tilde{R}$ among all such $R_{x}$. Then $\overline{K}$ slides freely in a radius $\tilde{R}$ ball.

Now suppose that a ball of radius $r$ rolls freely in $K$. Then at every boundary point $x$, a ball inside of $K$ intersects $K$ at $x$. The gnomonic projection of this ball is an ellipsoid in $\overline{K}$ for which the ratio of the axes is bounded. There is a maximal principal curvature $\kappa_{\max}$ among all these ellipsoids, and a ball of radius $1/\kappa_{\max}$ rolls freely in all of them. Consequently, it rolls freely in $\overline{K}$.

The hyperbolic volume of $K$ has the same distribution as the weighted volume of the convex hull of $n$ i.i.d. random points chosen by the probability density function $\psi/\int_{\overline{K}}\psi(x)\,\mathrm{d}x$ in $\mathbb{R}^{d}$, where $\psi=(1-\|x\|^{2})^{-(d+1)/2}$, $x\in\operatorname{int}B^{d}$ is the density of the image measure of $\operatorname{Vol}_{\mathbb{H}^{d}}$. For more information, see [BT20]*Section 5.2.

Thus, the conditions of Theorem 9 are satisfied, so we may apply its statement, which yields the desired asymptotic variance upper bound for $\operatorname{Vol}_{\mathbb{H}^{d}}(K_{n})$.

8. Direct proof of Theorem 5

Essentially the same way as in Section 3, the variance upper bound in Theorem 5 can be shown via a non-euclidean version of the economical cap theorem below.

Lemma 5**.**

The gnomonic projection preserves the order of magnitude of the volume of caps.

Proof.

Assume that $K\in\mathcal{K}(\mathbb{S}^{d})$ and $e_{d+1}$ is the center of the minimum radius ball (circumball) containing $K$. Note that the radius $R_{K}$ of the circumball is less than $\pi/2$. Consider a cap $C$ of $K$ with $\operatorname{Vol}_{\mathbb{S}^{d}}(C)=v$. Then

[TABLE]

where $u$ is the farthest point of $C$ from the center of projection, $e_{d+1}$. On the one hand, $V(g(C))>v$. On the other hand, the quantity $\frac{1}{\langle u,e_{d+1}\rangle^{d+1}}$ is bounded from above by a constant $c_{K,d}$ depending only on $K$ and $d$. Thus, $V(g(C))\leq c_{K,d}\cdot v$.

For the hyperbolic case, assume again that the center of the circumball of $K\in\mathcal{K}(\mathbb{H}^{d})$ is $e_{d+1}$. Let $D$ be a hyperbolic cap of $K$ with $\operatorname{Vol}_{\mathbb{H}^{d}}(D)=v$. Then $V(h(D))<v$, and $V(h(D))\geq v\cdot\frac{1}{\langle u,e_{d+1}\rangle^{d+1}}$, where $\frac{1}{\langle u,e_{d+1}\rangle^{d+1}}$ is bounded from below, similarly to the spherical case. ∎

Theorem 10**.**

Let $K\in\mathcal{K}(\mathbb{S}^{d})$ (or $\mathcal{K}(\mathbb{H}^{d})$, resp.) and $0<v<(2d)^{-2d}\cos^{d+1}R_{K}$ (or $0<v<(2d)^{-2d}$ resp.). Then there exist caps $C_{1},\dots,C_{m}$ and pairwise disjoint convex sets $C^{\prime}_{1},\dots,C^{\prime}_{m^{\prime}}$, with $m\approx m^{\prime}$, such that

1. (i)

$\bigcup_{i=1}^{m^{\prime}}C^{\prime}_{i}\subset K(v)\subset\bigcup_{i=1}^{m}C_{i}$, 2. (ii)

$\operatorname{Vol}_{\mathcal{M}^{d}}C_{i}\ll v$, $(i=1,\dots,m)$ and $\operatorname{Vol}_{\mathcal{M}^{d}}C^{\prime}_{i}\gg v$, $(i=1,\dots,m^{\prime})$, where $\mathcal{M}^{d}=\mathbb{S}^{d}$ or $\mathbb{H}^{d}$, 3. (iii)

for every cap $C$ of volume at most $v$, there is a $C_{i}$ containing $C$, 4. (iv)

every convex set $C^{\prime}_{i}$ is contained in some $C_{j}$.

Proof.

We only give the proof for the case of $\mathbb{S}^{d}$, the hyperbolic variant can be shown essentially in the same way. The wet part $K(v)$ is the union of caps $C_{\alpha}$ of spherical volume at most $v$. The images of the caps $C_{\alpha}$ under the gnomonic projection $g$ are the caps $\overline{C}_{\alpha}=g(C_{\alpha})$ in $\overline{K}=g(K)$ with possibly different volumes of $\bar{v}_{\alpha}$. By compactness, there exists a minimal and a maximal volume among $\bar{v}_{\alpha}$, let us denote these by $\bar{v}_{\min}$ and $\bar{v}_{\max}$, respectively. Consider the (Euclidean) wet parts in $\overline{K}$ with parameters $\bar{v}_{\min}$ and $\bar{v}_{\max}$. Then

[TABLE]

Apply Theorem 8 to both wet parts. Denote the caps and the convex sets contained in them provided by Theorem 8 corresponding to $\bar{v}_{\max}$ by $D_{i}$ and $D^{\prime}_{i}$ $(i=1,\dots,m)$ and those corresponding to $\bar{v}_{\min}$ by $E_{i}$ and $E^{\prime}_{i}$ $(i=1,\dots,m)$. Let $C_{i}=g^{-1}(D_{i})$, $(i=1,\dots,m)$ and $C^{\prime}_{i}=g^{-1}(E^{\prime}_{i})$ for some indices $i=1,\dots,m^{\prime}$ specified later.

The large caps $D_{i}$ cover $\overline{K}(\bar{v}_{\max})$ so they cover $g(K(v))$ as well. Furthermore, for any spherical cap $C$ of (spherical) volume at most $v$, its image $g(C)$ has volume at most $\bar{v}_{\max}$, therefore, it is contained in some $D_{i}$.

The sets $E^{\prime}_{i}$ are disjoint and contained in caps of volume $\bar{v}_{\min}$, therefore they are contained in $g(K(v))$, and there are caps $D_{j(i)}$ such that $E^{\prime}_{i}\subset D_{j(i)}$ for each $i\in\{1,\dots,m\}$. However, the same index $j$ can belong to multiple $i$’s, so some of the sets $E^{\prime}_{i}$ can be dropped to avoid multiplicity. The number of these sets cannot be too large due to the criteria on the volumes of $D_{i}$ and $E^{\prime}_{i}$. The preimages of the remaining sets will be the spherical sets $C^{\prime}_{i}$, $(i=1,\dots,m)$ with $m\approx m^{\prime}$.

Lemma 5 and the corresponding part of the Euclidean theorem yield (ii). ∎

9. Acknowledgements

The second author was supported by the Móricz Doctoral Scholarship.

The second author was also supported by the University Research Scholarship Programme (EKÖP) no. EKÖP-452-SZTE, which has been implemented with the support provided by the Ministry of Culture and Innovation of Hungary and the National Research, Development and Innovation Fund.

This research was supported by NKFIH project no. 150151, which has been implemented with the support provided by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation Fund, financed under the ADVANCED_24 funding scheme.

This research was supported by project TKP2021-NVA-09. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.